A294304 Sum of the ninth powers of the parts of the partitions of n into two distinct parts.

0, 0, 513, 19684, 282340, 2215782, 12313161, 52404624, 186884496, 572351860, 1574304985, 3922174980, 9092033028, 19656178794, 40357579185, 78666720832, 147520415296, 265720871304, 464467582161, 786155279940, 1299155279940, 2091077378830, 3300704544313

Offset

1,3

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-45,45,120,-120,-210,210,252,-252,-210,210,120,-120,-45,45,10,-10,-1,1).

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^9 + (n-i)^9.

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10).

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.

(End)

Mathematica

Table[Sum[i^9 + (n - i)^9, {i, Floor[(n-1)/2]}], {n, 30}]

Prog

(PARI) a(n) = sum(i=1, (n-1)\2, i^9 + (n-i)^9); \\ Michel Marcus, Nov 08 2017
(PARI) concat(vector(2), Vec(x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ Colin Barker, Nov 20 2017

Crossrefs

Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303.

Sequence in context: A351304 A017681 A013957 * A036087 A007487 A023878

Adjacent sequences: A294301 A294302 A294303 * A294305 A294306 A294307

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 27 2017

Status

approved